Question: Simplify and expand the following expression: $ \dfrac{2}{r + 1}+ \dfrac{4}{3r + 21}+ \dfrac{4r}{r^2 + 8r + 7} $
Explanation: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $3$ out of denominator in the second term: $ \dfrac{4}{3r + 21} = \dfrac{4}{3(r + 7)}$ We can factor the quadratic in the third term: $ \dfrac{4r}{r^2 + 8r + 7} = \dfrac{4r}{(r + 1)(r + 7)}$ Now we have: $ \dfrac{2}{r + 1}+ \dfrac{4}{3(r + 7)}+ \dfrac{4r}{(r + 1)(r + 7)} $ The least common multiple of the denominators is: $ (r + 1)(r + 7)$ In order to get the first term over $(r + 1)(r + 7)$ , multiply by $\dfrac{3(r + 7)}{3(r + 7)}$ $ \dfrac{2}{r + 1} \times \dfrac{3(r + 7)}{3(r + 7)} = \dfrac{6(r + 7)}{(r + 1)(r + 7)} $ In order to get the second term over $(r + 1)(r + 7)$ , multiply by $\dfrac{r + 1}{r + 1}$ $ \dfrac{4}{3(r + 7)} \times \dfrac{r + 1}{r + 1} = \dfrac{4(r + 1)}{(r + 1)(r + 7)} $ In order to get the third term over $(r + 1)(r + 7)$ , multiply by $\dfrac{3}{3}$ $ \dfrac{4r}{(r + 1)(r + 7)} \times \dfrac{3}{3} = \dfrac{12r}{(r + 1)(r + 7)} $ Now we have: $ \dfrac{6(r + 7)}{(r + 1)(r + 7)} + \dfrac{4(r + 1)}{(r + 1)(r + 7)} + \dfrac{12r}{(r + 1)(r + 7)} $ $ = \dfrac{ 6(r + 7) + 4(r + 1) + 12r} {(r + 1)(r + 7)} $ Expand: $ = \dfrac{6r + 42 + 4r + 4 + 12r}{3r^2 + 24r + 21} $ $ = \dfrac{22r + 46}{3r^2 + 24r + 21}$